<TeXmacs|2.1.1>

<style|<tuple|tmbook|chinese>>

<\body>
  <doc-data|<doc-title|\<#4E0D\>\<#7B49\>\<#5F0F\>\<#65E5\>\<#8BB0\>>|<doc-author|<author-data|<author-name|by
  zhcosin>>>>

  <chapter|2023>

  <section|2023-06>

  <\example>
    \<#8BBE\> <math|a,b,c\<geqslant\>0>\<#FF0C\>\<#8BC1\>\<#660E\>

    <\equation*>
      <frac|a<rsup|2>-b c|2a<rsup|2>+b<rsup|2>+c<rsup|2>>+<frac|b<rsup|2>-c
      a|2b<rsup|2>+c<rsup|2>+a<rsup|2>>+<frac|c<rsup|2>-a
      b|2c<rsup|2>+a<rsup|2>+b<rsup|2>>\<geqslant\>0
    </equation*>
  </example>

  <\proof>
    \<#5148\>\<#628A\>\<#51CF\>\<#53F7\>\<#5904\>\<#7406\>\<#6389\>

    <\equation*>
      <frac|a<rsup|2>-b c|2a<rsup|2>+b<rsup|2>+c<rsup|2>>=<frac|1|2>-<frac|<around*|(|b+c|)><rsup|2>|2<around*|(|2a<rsup|2>+b<rsup|2>+c<rsup|2>|)>>
    </equation*>

    \<#4E8E\>\<#662F\>\<#53EA\>\<#9700\>\<#8981\>\<#8BC1\>\<#660E\>

    <\equation*>
      <frac|<around*|(|b+c|)><rsup|2>|2a<rsup|2>+b<rsup|2>+c<rsup|2>>+<frac|<around*|(|c+a|)><rsup|2>|2b<rsup|2>+c<rsup|2>+a<rsup|2>>+<frac|<around*|(|a+b|)><rsup|2>|2c<rsup|2>+a<rsup|2>+b<rsup|2>>\<leqslant\>3
    </equation*>

    \<#7531\>\<#67EF\>\<#897F\>\<#4E0D\>\<#7B49\>\<#5F0F\>\<#FF0C\>\<#6709\>

    <\equation*>
      <frac|<around*|(|b+c|)><rsup|2>|2a<rsup|2>+b<rsup|2>+c<rsup|2>>\<leqslant\><frac|b<rsup|2>|a<rsup|2>+b<rsup|2>>+<frac|c<rsup|2>|a<rsup|2>+c<rsup|2>>
    </equation*>

    \<#53E6\>\<#5916\>\<#4E24\>\<#9879\>\<#540C\>\<#6B64\>\<#529E\>\<#7406\>\<#FF0C\>\<#6700\>\<#540E\>\<#4E09\>\<#5F0F\>\<#76F8\>\<#52A0\>\<#5373\>\<#8BC1\>.
  </proof>

  <\example>
    \<#5047\>\<#8BBE\> <math|x,y,z\<geqslant\>1>\<#FF0C\>\<#4E14\>
    <math|<frac|1|x>+<frac|1|y>+<frac|1|z>=2>\<#FF0C\>\<#8BC1\>\<#660E\>

    <\equation*>
      <sqrt|x+y+z>\<geqslant\><sqrt|x-1>+<sqrt|y-1>+<sqrt|z-1>
    </equation*>
  </example>

  <\proof>
    \<#8BB0\> <math|a=<sqrt|x-1>>,<math|b=<sqrt|y-1>>,<math|c=<sqrt|z-1>>\<#FF0C\>\<#5219\>\<#6761\>\<#4EF6\>\<#4E2D\>\<#7684\>\<#7B49\>\<#5F0F\>\<#5373\>\<#4E3A\>

    <\equation*>
      <frac|1|1+a<rsup|2>>+<frac|1|1+b<rsup|2>>+<frac|1|1+c<rsup|2>>=2
    </equation*>

    \<#8981\>\<#8BC1\>\<#660E\>\<#7684\>\<#4E0D\>\<#7B49\>\<#5F0F\>\<#5373\>\<#4E3A\>

    <\equation*>
      a<rsup|2>+b<rsup|2>+c<rsup|2>+3\<geqslant\><around*|(|a+b+c|)><rsup|2>
    </equation*>

    \<#6709\>

    <\equation*>
      <frac|a<rsup|2>|1+a<rsup|2>>+<frac|b<rsup|2>|1+b<rsup|2>>+<frac|c<rsup|2>|1+c<rsup|2>>=3-<around*|(|<frac|1|1+a<rsup|2>>+<frac|1|1+b<rsup|2>>+<frac|1|1+c<rsup|2>>|)>=1
    </equation*>

    \<#540C\>\<#65F6\>\<#7531\>\<#67EF\>\<#897F\>\<#4E0D\>\<#7B49\>\<#5F0F\>\<#5F97\>

    <\equation*>
      <frac|a<rsup|2>|1+a<rsup|2>>+<frac|b<rsup|2>|1+b<rsup|2>>+<frac|c<rsup|2>|1+c<rsup|2>>\<geqslant\><frac|<around*|(|a+b+c|)><rsup|2>|a<rsup|2>+b<rsup|2>+c<rsup|2>+3>
    </equation*>

    \<#5373\>\<#5F97\>\<#8BC1\>.

    \;
  </proof>

  <\example>
    \<#8BBE\> <math|a,b,c> \<#662F\>\<#4EFB\>\<#610F\>\<#4E09\>\<#4E2A\>\<#5B9E\>\<#6570\>\<#FF0C\>\<#8BB0\>

    <\equation*>
      x=<sqrt|b<rsup|2>-b c+c<rsup|2>>,y=<sqrt|c<rsup|2>-c
      a+a<rsup|2>>,z=<sqrt|a<rsup|2>-a b+b<rsup|2>>
    </equation*>

    \<#8BC1\>\<#660E\>

    <\equation*>
      x y+y z+z x\<geqslant\>a<rsup|2>+b<rsup|2>+c<rsup|2>
    </equation*>
  </example>

  <\proof>
    \<#6709\>

    <\equation*>
      x=<sqrt|<around*|(|b-<frac|c|2>|)><rsup|2>+<frac|3|4>c<rsup|2>>,y=<sqrt|<around*|(|a-<frac|c|2>|)><rsup|2>+<frac|3|4>c<rsup|2>>
    </equation*>

    \<#7531\>\<#67EF\>\<#897F\>\<#4E0D\>\<#7B49\>\<#5F0F\>\<#6709\>\ 

    <\equation*>
      x y=<sqrt|<around*|(|<around*|(|b-<frac|c|2>|)><rsup|2>+<frac|3|4>c<rsup|2>|)><around*|(|<around*|(|a-<frac|c|2>|)><rsup|2>+<frac|3|4>c<rsup|2>|)>>\<geqslant\><around*|(|a-<frac|c|2>|)><around*|(|b-<frac|c|2>|)>+<frac|3|4>c<rsup|2>=a
      b-<frac|1|2><around*|(|a+b|)>c+c<rsup|2>
    </equation*>

    \<#540C\>\<#7406\>\<#6709\>\<#5173\>\<#4E8E\> <math|y z> \<#548C\>
    <math|z x> \<#7684\>\<#53E6\>\<#5916\>\<#4E24\>\<#5F0F\>\<#FF0C\>\<#4E09\>\<#5F0F\>\<#76F8\>\<#52A0\>\<#5373\>\<#8BC1\>.

    \;
  </proof>

  <\example>
    \<#8BBE\> <math|a,b,c> \<#662F\>\<#4EFB\>\<#610F\>\<#4E09\>\<#4E2A\>\<#5B9E\>\<#6570\>\<#FF0C\>\<#8BB0\>

    <\equation*>
      x=<sqrt|b<rsup|2>+b c+c<rsup|2>>,y=<sqrt|c<rsup|2>+c
      a+a<rsup|2>>,z=<sqrt|a<rsup|2>+a b+b<rsup|2>>
    </equation*>

    \<#8BC1\>\<#660E\>

    <\equation*>
      x y+y z+z x\<geqslant\><around*|(|a+b+c|)><rsup|2>
    </equation*>
  </example>

  <\proof>
    \<#4EFF\>\<#4E0A\>\<#4F8B\>\<#6709\>

    <\equation*>
      x=<sqrt|<around*|(|b+<frac|c|2>|)><rsup|2>+<frac|3|4>c<rsup|2>>,y=<sqrt|<around*|(|a+<frac|c|2>|)><rsup|2>+<frac|3|4>c<rsup|2>>
    </equation*>

    \<#4ECD\>\<#7531\>\<#67EF\>\<#897F\>\<#6709\>

    <\equation*>
      x y=<sqrt|<around*|(|<around*|(|b+<frac|c|2>|)><rsup|2>+<frac|3|4>c<rsup|2>|)><around*|(|<around*|(|a+<frac|c|2>|)><rsup|2>+<frac|3|4>c<rsup|2>|)>>\<geqslant\><around*|(|a+<frac|c|2>|)><around*|(|b+<frac|c|2>|)>+<frac|3|4>c<rsup|2>=a
      b+<frac|1|2><around*|(|a+b|)>c+c<rsup|2>
    </equation*>

    \<#5BF9\> <math|y z> \<#548C\> <math|z x>
    \<#4EFF\>\<#6B64\>\<#5904\>\<#7406\>\<#FF0C\>\<#518D\>\<#5C06\>\<#6700\>\<#540E\>\<#5F97\>\<#5230\>\<#7684\>\<#4E09\>\<#5F0F\>\<#76F8\>\<#52A0\>\<#5373\>\<#8BC1\>.

    \;
  </proof>

  \;

  \;
</body>

<\initial>
  <\collection>
    <associate|page-medium|paper>
  </collection>
</initial>

<\references>
  <\collection>
    <associate|auto-1|<tuple|1|5>>
    <associate|auto-2|<tuple|1.1|5>>
  </collection>
</references>

<\auxiliary>
  <\collection>
    <\associate|toc>
      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|font-shape|<quote|small-caps>|1.<space|2spc>2023>
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <pageref|auto-1><vspace|0.5fn>

      1.1.<space|2spc>2023-06 <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-2>
    </associate>
  </collection>
</auxiliary>